Newton's Law of Cooling

Temperature vs Time T(t)

Temperature T(t) Environment Temperature Current Temperature

Object Visualization

Current Values

Current Time 0.00 s

Current Temperature 0.00 °C

Cooling Rate 0.00 °C/s

Temperature Difference 0.00 °C

Time Constant 0.00 s

Equilibrium Progress 0.00%

Current Values

Cooling Parameters

Initial Temperature T₀: 80 °C Environment Temperature: 20 °C Cooling Constant k: 0.05 s⁻¹

Time Range

Maximum Time: 100 s Animation Speed: 1.0x

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Newton's Law of Cooling Equations

Newton's Law of Cooling: T(t) = T_env + (T₀ - T_env)·e^(-kt)

Cooling Constant k k = h·A/(m·c)

Time Constant τ = 1/k (time to reach 63.2% of equilibrium)

Rate of Temperature Change: dT/dt = -k(T - T_env)

What is Newton's Law of Cooling?

Newton's Law of Cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. This law, formulated by Isaac Newton in 1701, describes how objects cool down or warm up toward thermal equilibrium with their environment. The temperature follows an exponential decay curve, approaching but never quite reaching the environmental temperature.

Physical Meaning

Cooling Constant (k): Determines how quickly the object cools. It depends on surface area (A), heat transfer coefficient (h), mass (m), and specific heat capacity (c). Larger k means faster cooling.
Time Constant (τ = 1/k): The time required for the temperature difference to decrease to about 36.8% (1/e) of its initial value. After 5τ, the object is within 1% of equilibrium.
Exponential Decay: The temperature difference decreases exponentially: ΔT(t) = ΔT₀·e^(-kt). The rate of cooling is proportional to how far the object is from equilibrium.

Factors Affecting Cooling Rate

Surface Area: Larger surface area increases heat transfer, increasing k.
Heat Transfer Coefficient (h): Depends on convection, conduction, and radiation. Air has lower h than water.
Mass and Specific Heat: Larger mass or higher specific heat means more thermal energy, decreasing k.
Temperature Difference: Larger difference initially causes faster cooling, slowing as equilibrium approaches.

Real-World Applications

Food and Beverage: Predicting cooling times for hot drinks, food safety temperature monitoring, refrigeration design.
Forensic Science: Estimating time of death from body temperature (algor mortis), using cooling curves to determine death timing.
Engineering: Heat exchanger design, electronic cooling systems, material processing, HVAC systems.
Medicine: Therapeutic hypothermia, fever monitoring, cryotherapy temperature control.

Limitations

The law assumes constant environmental temperature and uniform object temperature. It works well for: convection-dominated cooling, small temperature differences, and when internal heat transfer is fast. For large temperature differences, radiative cooling (T⁴ dependence) becomes significant. Phase changes (melting/freezing) violate the assumptions as latent heat is released/absorbed.